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The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra?

Principal manuscript, Theorem 1.1: (i) a finite lattice is representable iff it admits a finite colored-graph witness; (ii) no Turing machine decides representability from the order table, so not every finite lattice is representable; (iii) there is a nonrepresentable lattice of minimum size, larger than 7, specified by a finite description involving a Busy Beaver constant (its size is not computed); (iv) recognizing full subgroup intervals of finite groups is undecidable. The companion constructs a finite lattice that is no interval $[D,G]$ in a finite group, hence (Palfy-Pudlak) a nonrepresentable one. No explicit small counterexample is exhibited.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra? · Finite lattice representation problem

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VibeMathed record: The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and Theorems 1.1-1.2 of the companion were read against the problem; both claim a finite lattice that is not the full congruence lattice of any finite algebra, the companion via a lattice that is no finite subgroup interval. Proofs (over 170 and 80 pages, using the classification of finite simple groups and Larsen-Pink) were not refereed. A Lean challenge exists, lean/ComparatorChallenges/FiniteCongruenceGraph.json (solution module OAI.Algebra.Universal.GraphCriterion present), not in the formalization catalogue; its statement, read here, is only the colored-graph criterion (a finite lattice is representable iff it has a finite colored-graph witness). That criterion does not imply the negative answer, so the entry is left unreviewed. Not rebuilt here.